Ionospheric correction for single frequency GPS receivers using three satellites

ABSTRACT

Systems and methods are disclosed for ionospheric correction in a system employing a single GPS frequency receiver for determining the geographic location of an object on the earth&#39;s surface. The receiver receives signals transmitted at the GPS L 1  frequency from at least first, second and third GPS satellites, the first, second and third satellites having respective orbital positions relative to the receiver such that the total electron count (TEC) contribution to ionoshperic interference to signals transmitted by the respective satellites and received by the receiver is approximately the same. Respective measured distances of the three satellites to the receiver are determined based on the actual signal transmission times. True distances of the respective satellites are then calculated based on the assumption that the TEC contribution to the interference from each satellite is approximately the same.

FIELD OF THE INVENTION

The present invention pertains generally to the field of locatingsystems employing GPS satellites, and more specifically to methods andapparatus for ionospheric correction in a locating system employing asingle frequency GPS receiver.

BACKGROUND OF THE INVENTION

Satellite-based global positioning systems are well known. For example,U.S. Pat. No. 5,210,540 (the “'540 patent”), issued to Masumotadiscloses a global positioning system for locating a mobile object, suchas an automobile, in a global geometrical region. As described therein,the system includes a Global Positioning System (GPS) receiver forreceiving radio waves from a plurality of satellites and outputtingeither two or three dimensional position data indicative of the presentposition of the mobile object. For greater accuracy, an altimeter isalso employed to detect the mobile object's relative altitude. U.S. Pat.No. 5,430,654 (the “'654 patent”), issued to Kyrtsos et al disclosesapparatus and methods for determining the position of a vehicle at, ornear the surface of the earth using a satellite-based navigation system,wherein “precise” position estimates are achieved by reducing theeffective receiver noise. The respective '540 and '654 patents areincorporated herein by reference for all they disclose and teach.

In particular, the Global Positioning System is a satellite-basednavigation system that was designed and paid for by the U.S. Departmentof Defense. The GPS consists of twenty-four satellites, which orbit theearth at known coordinates. The particulars of the GPS are described insection 3.4.2 of “Vehicle Location and Navigation Systems,” by YilinZhao, Artech House, Inc., 1997, which is fully incorporated herein byreference. As noted therein, the observation of at least four GPSsatellites simultaneously will permit determination of three-dimensionalcoordinates of a receiver located on the earth's surface, as well as thetime offset between the receiver and the respective satellites.

One problem encountered in global positioning systems is ionosphericinterference. The ionosphere is a dispersive medium, which lies betweenseventy and one thousand kilometers above the earth's surface. Theionosphere effects a certain, frequency dependent propagation delay onsignals transmitted from the respective GPS satellites. The ionospherealso effects GPS signal tracking by the receiver. Notably, theionospheric delay of a transmitted GPS signal can cause an error of upto ten meters when calculating the exact geographic position on theearth's surface of the receiver.

As demonstrated below, delay from ionospheric interference can be almostcompletely corrected for by using multiple frequency observations, i.e.,by transmitting and receiving signals at two different, known GPSfrequencies, L₁ and L₂, from a respective satellite. However, forsecurity reasons, most GPS receivers do not receive the L₂ frequency.Instead, these single (i.e., L₁) frequency receivers can employ a modelto estimate and correct for transmission delay due to ionosphericinterference.

For example, the Global Positioning System, Interface Control Document,ICD-GPS-200, Revision C, Initial Release, Oct. 10, 1993, provides amethod for ionospheric correction based on an “approximate atmospheric”model, which is dependent on a “total electron content” (TEC) model. Inaccordance with this model, and using only the L₁ signal transmissionfrequency, it can be shown that the error ΔS₁ in the true satellite toreceiver/user distance ρ is: $\begin{matrix}{{\Delta \quad S_{1}} = {F_{pp}40.3\frac{TEC}{L_{1}^{2}}}} & (1)\end{matrix}$

where F_(pp), is an “obliquity factor”: $\begin{matrix}{F_{pp} = \left( \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi}{R_{e} + H_{r}} \right)^{2}} \right\rbrack \right)^{- \frac{1}{2}}} & (2)\end{matrix}$

where R_(e) is the radius of the earth, H_(r) is the height of themaximum electron density in the ionosphere from the earth's surface, andφ is the angle between the respective satellite and a plane tangent tothe earth's surface at the receiver's position.

Notably, the true TEC value of the ionosphere is very difficult to modeland is highly sensitive to variables, such as time of day, solaractivity and relative incident angle of the satellite with respect tothe sunlight trajectory (if any) toward the receiver location, etc. Inparticular, the TEC nominal value varies widely, within a range ofbetween 10¹⁶ to 10¹⁹. As a result, the above ionospheric correctionmodel has been shown to adequately correct for no more than 50% of theionospheric transmission delay.

As noted above, a dual frequency receiver can virtually eliminateionospheric errors by computing the pseudo-range distance of therespective satellite on both the L₁ and L₂ frequencies. For purposes ofillustration, a short derivation of such a dual frequency correctionmethodology is as follows:

Let ΔS₁ and ΔS₂ represent the error in the pseudo-range distancescomputed at frequencies L₁ and L₂, respectively. Then: $\begin{matrix}{{\Delta \quad S_{1}} = {{{- F_{pp}}40.3\frac{TEC}{L_{1}^{2}}} = {\lambda_{L_{1}} - \rho}}} & (3)\end{matrix}$

and $\begin{matrix}{{\Delta \quad S_{2}} = {{{- F_{pp}}40.3\frac{TEC}{L_{2}^{2}}} = {\lambda_{L2} - \rho}}} & (4)\end{matrix}$

where λ_(L1) and λ_(L2) are the respective pseudo-range distancescomputed at frequencies L₁ and L₂, respectively, and ρ is the truesatellite to receiver distance. Dividing equation (3) by equation (4),results in: $\begin{matrix}{\frac{\Delta \quad S_{1}}{\Delta \quad S_{2}} = \frac{L_{2}^{2}}{L_{1}^{2}}} & (5)\end{matrix}$

Subtracting equation (4) from equation (3), gives:

ΔS ₁ −ΔS ₂=λ_(L1)−λ_(L2)   (6)

Substituting equation (5) into equation (6), and after some minoralgebraic manipulation, provides: $\begin{matrix}{{\Delta \quad S_{1}} = {\left( \frac{L_{2}^{2}}{L_{1}^{2} - L_{1}^{2}} \right)\quad \left( {\lambda_{L_{2}} - \lambda_{L_{1}}} \right)}} & (7)\end{matrix}$

Importantly, all quantities in the above expression (7) are either knownby the receiver, or can be measured, with the TEC value totally canceledout of the equation.

Of course, in a GPS-based locating system having only a single (i.e.,L₁) frequency receiver, the above-described ionospheric correction modelbased on both the L₁ and L₂ transmission frequencies can not beemployed.

SUMMARY OF THE INVENTION

The present invention is directed to satellite-based systems and methodsfor correcting for ionospheric interference in a single-frequencyreceiver system for determining the geographic location of an object onthe earth's surface without requiring ionospheric TEC modeling.

In accordance with one aspect of the invention, a method using satellitesignal transmission for determining the geographic location of areceiver on the earth's surface, includes:

receiving a first signal transmitted at a known frequency from a firstsatellite having a known orbital position;

receiving a second signal transmitted at the same frequency as the firstsignal from a second satellite having a known orbital position;

receiving a third signal transmitted at the same frequency as the firstsignal from a third satellite having a known orbital position;

calculating measured distances λ¹, λ² and λ³ of the respective first,second and third satellites from the receiver based at least in part onthe transmission time of the third signal; and

calculating actual distances ρ¹, ρ² and ρ³ of the respective first,second and third satellites from the receiver based on the measureddistances λ¹, λ² and λ³, according to the relationships${{\frac{F_{pp}^{1}}{F_{pp}^{2}}\left( {\lambda^{2} - \rho^{2}} \right)} = {\lambda^{1} - \rho^{1}}},{{\frac{F_{pp}^{1}}{F_{pp}^{3}}\left( {\lambda^{3} - \rho^{3}} \right)} = {\lambda^{1} - \rho^{1}}},{and}$${{\frac{F_{pp}^{2}}{F_{pp}^{3}}\left( {\lambda^{3} - \rho^{3}} \right)} = {\lambda^{2} - \rho^{2}}},$

where F_(pp) ¹, F_(pp) ², and F_(pp) ³ are obliquity factors for therespective first, second and third satellites based on the respectiveangles φ₁, φ₂ and φ₃ they form with a plane tangent to the earth'ssurface at the geographic location of the receiver, with${F_{pp}^{1} = {- \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi_{1}}{R_{e} + H_{r}} \right)^{2}} \right\rbrack^{- \frac{1}{2}}}},{F_{pp}^{2} = {- \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi_{2}}{R_{e} + H_{r}} \right)^{2}} \right\rbrack^{- \frac{1}{2}}}},{and}$${F_{pp}^{3} = \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi_{3}}{R_{e} + H_{r}} \right)^{2}} \right\rbrack^{- \frac{1}{2}}},$

where R_(e) is the radius of the earth and H_(r) is the height ofmaximum electron density in the ionosphere surrounding the earth'ssurface.

As will be apparent to those skilled in the art, other and furtheraspects and advantages of the present invention will appear hereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

Preferred embodiments of the present invention are illustrated by way ofexample, and not by way of limitation, in the accompanying drawing, inwhich:

FIG. 1 is a diagrammatic illustration of a GPS-based locating system forlocating the geographic position of a single frequency GPS receiver onthe earth's surface.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

An exemplary GPS-based locating system is illustrated in FIG. 1. Forease in illustration of the present inventions disclosed and describedherein, only three satellites and a single receiver are shown in thelocating system. It is assumed that the timing offset between therespective satellites and receiver is either already known or otherwisecalculated by use of an additional satellite (not shown).

In particular, a GPS receiver 50 is at a particular location on thesurface 22 of the earth 24, e.g., within an object to be located. Afirst GPS satellite 56 is located above the earth's atmosphere at anangle φ₁ between the satellite 56 and a plane 52 tangent to the earth'ssurface 22 at the geographic location of the GPS receiver 50. The actualdistance between the first satellite 56 and the receiver 50 isdesignated as ρ^(GPS1). Likewise, a second GPS satellite 57 is locatedabove the earth's atmosphere at an angle φ₂ formed between satellite 57and plane 52, with the actual distance between the second satellite 57and the receiver 50 designated as ρ^(GPS2). A third GPS satellite 58 islocated above the earth's atmosphere at an angle φ₃ formed betweensatellite 58 and plane 52, with the actual distance between the thirdsatellite 58 and the receiver 50 designated as ρ^(GPS3). Although theGPS satellites 56, 57 and 58 broadcast at both the L_(1 and L) ₂frequencies, in accordance with the inventions disclosed herein, it isassumed that only the L₁ frequency is received by the receiver 50.Notably, the respective L₁ signals travel through the earth's ionosphere32 before they reach the GPS receiver 50.

More particularly, the ionospheric TEC is mainly a function of the timeof day, solar activities, and relative position of the satelliterelative to the respective sunlight trajectory (if any) towards thereceiver 50. Thus, at a given instant while satellites 56, 57 and 58 arein orbit over the receiver 50, the ionospheric TEC value for eachsatellite is approximately the same. Based on signal transmissionmeasurements taken from each satellite 56, 57 and 58 at the same time ofday and, thus, with the same solar activity, the only variable is theparticular relative location of each satellite with respect to thereceiver 50.

In a preferred embodiment, the respective satellites 56, 57 and 58 areselected for making a location calculation by the receiver 50 based ontheir relative proximity in space to each other, thereby increasing thelikelihood that the respective nominal TEC values for each satellite canbe assumed to be approximately the same.

Accordingly, from above equation (1), the respective errors due toionospheric interference in the true satellite to receiver distancesρ^(GPS1), ρ^(GPS2) and ρ^(GPS3) are as follows: $\begin{matrix}{{\Delta \quad S_{1}} = {{{- F_{pp}^{GPS1}}40.3\frac{TEC}{L_{1}^{2}}} = {\lambda_{L1}^{GPS1} - \rho^{GPS1}}}} & (8)\end{matrix}$

$\begin{matrix}{{\Delta \quad S_{2}} = {{{- F_{pp}^{GPS2}}40.3\frac{TEC}{L_{1}^{2}}} = {\lambda_{L_{1}}^{GPS2} - \rho^{GPS2}}}} & (9)\end{matrix}$

and $\begin{matrix}{{\Delta \quad S_{3}} = {{{- F_{pp}^{GPS3}}40.3\frac{TEC}{L_{1}^{2}}} = {\lambda_{L_{1}}^{GPS3} - \rho^{GPS3}}}} & (10)\end{matrix}$

where λ_(L1) ^(GPS1), λ_(L1) ^(GPS2), λ₁ ^(GPS3) are the measuredpseudo-range distances between the respective satellites 56, 57 and 58and the receiver 50, and where F_(pp) ^(GPS1), F_(pp) ^(GPS2), andF_(pp) ^(GPS3) are the obliquity factors for satellites 56, 57 and 58based on the respective angles φ₁, φ₂ and φ₃ they form with the tangentplane 52. Following from equation (2): $\begin{matrix}{F_{pp}^{GPS1} = {- \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi_{1}}{R_{e} + H_{r}} \right)^{2}} \right\rbrack^{- \frac{1}{2}}}} & (11)\end{matrix}$

$\begin{matrix}{F_{pp}^{GPS2} = {- \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi_{2}}{R_{e} + H_{r}} \right)^{2}} \right\rbrack^{- \frac{1}{2}}}} & (12)\end{matrix}$

and $\begin{matrix}{F_{pp}^{GPS3} = \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi_{3}}{R_{e} + H_{r}} \right)^{2}} \right\rbrack^{- \frac{1}{2}}} & (13)\end{matrix}$

where R_(e) is the radius of the earth, H_(r) is the height of themaximum electron density in the ionosphere from the earth's surface,which are known to the receiver 50.

Considering the relationship between the first two satellites 56 and 57,and dividing equation (8) by equation (9): $\begin{matrix}{\frac{F_{pp}^{GPS1}}{F_{pp}^{GPS2}} = \frac{\lambda_{L1}^{GPS1} - \rho^{GPS1}}{\lambda_{L1}^{GPS2} - \rho^{GPS2}}} & (14)\end{matrix}$

Rearranging: $\begin{matrix}{{\frac{F_{pp}^{GPS1}}{F_{pp}^{GPS2}}\left( {\lambda_{L_{1}}^{GPS2} - \rho^{GPS2}} \right)} = {\lambda_{L_{1}}^{GPS1} - \rho^{GPS1}}} & (15)\end{matrix}$

By the same approach, it follows that: $\begin{matrix}{{\frac{F_{pp}^{GPS1}}{F_{pp}^{GPS3}}\left( {\lambda_{L_{1}}^{GPS3} - \rho^{GPS3}} \right)} = {\lambda_{L_{1}}^{GPS1} - \rho^{GPS1}}} & (16)\end{matrix}$

and $\begin{matrix}{{\frac{F_{pp}^{GPS2}}{F_{pp}^{GPS3}}\left( {\lambda_{L_{1}}^{GPS3} - \rho^{GPS3}} \right)} = {\lambda_{L_{1}}^{GPS2} - \rho^{GPS2}}} & (17)\end{matrix}$

Thus, the three equations (15), (16) and (17) contain only threeunknowns ρ^(GPS1), ρ^(GPS2) and ρ^(GPS3), whereby the true geographiclocation on the earth's surface 22 of the receiver 50 may be readilydetermined from the respective measured pseudo range distances of thethree satellites 56, 57 and 58 at the receiver 50. Notably, the solutionwill not depend on the respective TEC values.

Alternatively, correction for ionospheric interference may be done bymeasuring signals from just two satellites, e.g., 56 and 57.

Because the exact orbit position coordinates of the satellites 56 and 57(and, thus φ₁, and φ₂) are known to the receiver 50, the actual distanceρ^(GPS1-2) between the two satellites can be calculated as follows:$\begin{matrix}{\rho^{{GPS1} - 2} = \sqrt{\left( {x^{GPS1} - x^{GPS2}} \right)^{2} + \left( {y^{GPS1} - y^{GPS2}} \right)^{2} + \left( {z^{GPS1} - z^{GPS2}} \right)^{2}}} & (19)\end{matrix}$

where (x^(GPS1), y^(GPS1), z^(GPS1)) are the coordinates in space ofsatellite 56, and (x^(GPS2), y^(GPS2), z^(GPS2)) are the coordinates inspace of satellite 57.

Referring again to FIG. 2, and considering the triangle formed byρ^(GPS1-2), ρ^(GPS1) and ρ^(GPS2):

(ρ^(GPS1-2))²=(ρ^(GPS1))²+(ρ^(GPS2))²−2·ρ^(GPS1)·ρ^(GPS2)·cos α  (20)

where α is the angle between ρ^(GPS1) and ρ^(GPS2).

Letting β be the “measured angle” between λ_(L1) ^(GPS1) and λ_(L1)^(GPS2), since λ_(L1) ^(GPS1)−ρ^(GPS1)<<λ_(L1) ^(GPS1) and λ_(L1)^(GPS2)−ρ^(GPS2)<<λ_(L1) ^(GPS2), it is reasonable to assume cosα isapproximately equal to cosβ. Therefore, equation (20) can be rewrittenas

(ρ^(GPS1-2))²=(ρ^(GPS1))²+(ρ^(GPS2))²−2·ρ^(GPS1)·ρ^(GPS2)·cos β  (21)

Thus, the two equations, (21) and (15) contain only two unknowns,ρ^(GPS1) and ρ^(GPS2), whereby the true geographic location on theearth's surface 22 of the receiver 50 may be readily determined from therespective measured pseudo range distances λ_(L1) ^(GPS1) and λ_(L1)^(GPS2) of satellites 56 and 57, and, thus, the measured angle Ø formedby satellites 56 and 57 with respect to the receiver 50. Again, thesolution will not depend on the respective TEC values.

As will be apparent to those skilled in the art, the above methods forionospheric correction may also be applied to a GLONASS, or combinedGPS/GLONASS receiver, so long as the respective transmission timing,frequency and orbit coordinates are known by the receiver for eachrespective satellite.

While preferred systems and methods for correcting for ionosphericinterference in a single frequency GPS receiver system have been shownand described, as would be apparent to those skilled in the art, manymodifications and applications are possible without departing from theinventive concepts herein. Thus, the scope of the disclosed invention isnot to be restricted except in accordance with the appended claims.

What is claimed is:
 1. A method using satellite signal transmission fordetermining the geographic location of a receiver on the earth'ssurface, comprising: receiving a first signal transmitted at a knownfrequency from a first satellite having a known orbital position;receiving a second signal transmitted at the same frequency as the firstsignal from a second satellite having a known orbital position;receiving a third signal transmitted at the same frequency as the firstsignal from a third satellite having a known orbital position;calculating measured distances λ¹, λ² and λ³ of the respective first,second and third satellites from the receiver based at least in part onthe transmission time of the respective first, second and third signals;and calculating actual distances ρ¹, ρ² and ρ³ of the respective first,second and third satellites from the receiver based on the measureddistances λ¹, λ² and λ³, according to the relationships${{\frac{F_{pp}^{1}}{F_{pp}^{2}}\left( {\lambda^{2} - \rho^{2}} \right)} = {\lambda^{1} - \rho^{1}}},{{\frac{F_{pp}^{1}}{F_{pp}^{3}}\left( {\lambda^{3} - \rho^{3}} \right)} = {\lambda^{1} - \rho^{1}}},{and}$${{\frac{F_{pp}^{2}}{F_{pp}^{3}}\left( {\lambda^{3} - \rho^{3}} \right)} = {\lambda^{2} - \rho^{2}}},$

where F_(pp) ¹, F_(pp) ², and F_(pp) ³ are obliquity factors for therespective first, second and third satellites based on the respectiveangles φ₁, φ₂ and φ₃ they form with a plane tangent to the earth'ssurface at the geographic location of the receiver, with${F_{pp}^{1} = {- \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi_{1}}{R_{e} + H_{r}} \right)^{2}} \right\rbrack^{- \frac{1}{2}}}},{F_{pp}^{2} = {- \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi_{2}}{R_{e} + H_{r}} \right)^{2}} \right\rbrack^{- \frac{1}{2}}}},{and}$${F_{pp}^{3} = \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi_{3}}{R_{e} + H_{r}} \right)^{2}} \right\rbrack^{- \frac{1}{2}}},$

where R_(e) is the radius of the earth and H_(r) is the height ofmaximum electron density in the ionosphere surrounding the earth'ssurface.
 2. The method of claim 1, wherein the first second and thirdsatellites are global positioning system (GPS) satellites.
 3. The methodof claim 2, wherein the first second and third signals are transmittedat the GPS L₁ signal frequency.
 4. The method of claim 1, wherein thefirst, second and third satellites have respective orbital positionsrelative to the receiver such that the total electron count (TEC)contribution to ionoshperic interference to the transmission of thefirst, second and third signals is approximately the same.
 5. A systemfor determining the geographic location of objects on the earth'ssurface, comprising: a receiver configured to receive signalstransmitted at a selected frequency from first, second and thirdsatellites having known orbital positions, the receiver furtherconfigured to calculate measured distances λ¹, λ² and λ³ of therespective first, second and third satellites from the receiver based atleast in part on the transmission time of the received signals, andcalculate actual distances ρ¹, ρ² and ρ³ of the respective first, secondand third satellites from the receiver based on the measured distancesλ¹, λ² and λ³, according to the relationships${{\frac{F_{pp}^{1}}{F_{pp}^{2}}\left( {\lambda^{2} - \rho^{2}} \right)} = {\lambda^{1} - \rho^{1}}},{{\frac{F_{pp}^{1}}{F_{pp}^{3}}\left( {\lambda^{3} - \rho^{3}} \right)} = {\lambda^{1} - \rho^{1}}},{and}$${{\frac{F_{pp}^{2}}{F_{pp}^{3}}\left( {\lambda^{3} - \rho^{3}} \right)} = {\lambda^{2} - \rho^{2}}},$

where F_(pp) ¹, F_(pp) ², and F_(pp) ³ are obliquity factors for therespective first, second and third satellites based on the respectiveangles φ₁, φ₂ and φ₃ they form with a plane tangent to the earth'ssurface at the geographic location of the receiver, with${F_{pp}^{1} = {- \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi_{1}}{R_{e} + H_{r}} \right)^{2}} \right\rbrack^{- \frac{1}{2}}}},{F_{pp}^{2} = {- \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi_{2}}{R_{e} + H_{r}} \right)^{2}} \right\rbrack^{- \frac{1}{2}}}},{and}$${F_{pp}^{3} = \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi_{3}}{R_{e} + H_{r}} \right)^{2}} \right\rbrack^{- \frac{1}{2}}},$

where R_(e) is the radius of the earth and H_(r) is the height ofmaximum electron density in the ionosphere surrounding the earth'ssurface.
 6. The system of claim 5, wherein the first second and thirdsatellites are global positioning system (GPS) satellites.
 7. The systemof claim 6, wherein the first second and third signals are transmittedat the GPS L₁ signal frequency.
 8. The system of claim 5, wherein thefirst, second and third satellites have respective orbital positionsrelative to the receiver such that the total electron count (TEC)contribution to ionoshperic interference to the transmission of thefirst, second and third signals is approximately the same.
 9. A systemfor determining the geographic location of objects on the earth'ssurface, comprising: a single frequency global positioning system (GPS)receiver configured to receive signals transmitted at the GPS L₁frequency from first, second and third GPS satellites, the respectivereceiver and satellites having synchronized clocks, the first, secondand third satellites having respective orbital positions relative to thereceiver such that the total electron count (TEC) contribution toionoshperic interference to signals transmitted by the respectivesatellites and received by the receiver is approximately the same, thereceiver further configured to calculate measured distances λ¹, λ² andλ³ of the respective first, second and third satellites from thereceiver based at least in part on the transmission time of the receivedsignals, and calculate actual distances ρ^(GPS1), ρ^(GPS2) and ρ^(GPS3)of the respective first, second and third satellites from the receiverbased on the measured distances λ_(L1) ^(GPS1), λ_(L1) ^(GPS2) andλ_(L1) ^(GPS3) according to the relationships${{\frac{F_{pp}^{GPS1}}{F_{pp}^{GPS2}}\left( {\lambda_{L_{1}}^{GPS2} - \rho^{GPS2}} \right)} = {\lambda_{L_{1}}^{GPS1} - \rho^{GPS1}}},{{\frac{F_{pp}^{GPS1}}{F_{pp}^{GPS3}}\left( {\lambda_{L_{1}}^{GPS3} - \rho^{GPS3}} \right)} = {\lambda_{L_{1}}^{GPS1} - \rho^{GPS1}}},{and}$${{\frac{F_{pp}^{GPS2}}{F_{pp}^{GPS3}}\left( {\lambda_{L_{1}}^{GPS3} - \rho^{GPS3}} \right)} = {\lambda_{L_{1}}^{GPS2} - \rho^{GPS2}}},$

where F_(pp) ^(GPS1), F_(pp) ^(GPS2), and F_(pp) ^(GPS3) are obliquityfactors for the respective first, second and third satellites based onthe respective angles φ₁, φ₂ and φ₃ they form with a plane tangent tothe earth's surface at the geographic location of the receiver, with${F_{pp}^{GPS1} = {- \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi_{1}}{R_{e} + H_{r}} \right)^{2}} \right\rbrack^{- \frac{1}{2}}}},{F_{pp}^{GPS2} = {- \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi_{2}}{R_{e} + H_{r}} \right)^{2}} \right\rbrack^{- \frac{1}{2}}}},{and}$${F_{pp}^{GPS3} = \left\lbrack {1 - \left( \frac{R_{e}\cos \quad \varphi_{3}}{R_{e} + H_{r}} \right)^{2}} \right\rbrack^{- \frac{1}{2}}},$

where R_(e) is the radius of the earth and H_(r) is the height ofmaximum electron density in the ionosphere surrounding the earth'ssurface.